428 Asian Journal of Control, Vol. 6, No. 3, pp. 428-431, September 2004

Manuscript received December 6, 2002, revised March 11,2003, accepted September 1, 2003.

The authors are with Department of Electrical andComputer Engineering, National University of Singapore, 10Kent Ridge Crescent, Singapore 119260.

－Brief Paper－

EXISTENCE ANALYSIS FOR LIMIT CYCLES OF RELAY FEEDBACK SYSTEMS

Chong Lin, Qing-Guo Wang, and Tong Heng Lee

ABSTRACT

This paper is concerned with the problems on the existence of limit cycles for SISO linear systems under relay feedback. A sufficient condition is given based on the Brouwer’s fixed point theorem.

KeyWords: Relay feedback systems, limit cycles, Brouwer’s fixed point theorem.

I. INTRODUCTION

Relay feedback systems have been widely employed in a broad technological domain for many decades [1]. Such a system may exhibit some special behaviours such as non-uniqueness of solutions [2,3], fast switches or chattering [4,5], sliding motion [6] and chaos [7]. An important property is that the system trajectories may tend to a periodic orbit. This periodic orbit is often termed a limit cycle in the literature. This property is very useful in modern control applications such as automatic tuning of controllers and frequency response estimation and identification [8~11]. Such practical applications activate the intensive investigation for limit cycle behaviors. However, most analysis work have to be based on the assumption that a limit cycle does exist, due to the difficulty of determining if it is really the case. Many classical results based on the describing function method are surveyed in [1]. The describing function method gives an approximate analysis for limit cycle behaviors for SISO systems. An exact approach is reported in [12] and a necessary condition is given. The drawback is to compute a period parameter through numerical method and to determine it really corresponds to a limit cycle. Another work [4] considers the normalized state-space realization from a given transfer function, and presents a sufficient

condition for the existence of a symmetric stable limit cycle with chattering.

In this paper, we will study the problem of existence of limit cycles for relay feedback systems with hysteresis and time delay. We will present a sufficient condition by using the Brouwer’s fixed point theorem. This paper is organized as follows. In Section 2, the considered system and problem are formulated. Section 3 establishes some useful lemmas and gives the main result for the existence problem. This paper is concluded in Section 4.

Notation: and n

denote, respectively, the field of real numbers and the n-dimensional real Euclidean space; I is the identity matrix; the superscript ‘T ’ stands for the matrix transpose; || ⋅ || denotes the Euclidean norm for a vector, or the spectral norm for a matrix; f (t−) : = lim∈ → 0 + f (t−∈)

II. PROBLEM FORMULATION

Consider a single-input single-output relay feed- back system described by

( ) ( ) ( )x t Ax t bu t τ= + −

( ) ( )y t cx t= (1)

( ) sgn ( ( ))du t y t= − ,

where ( ) ,nx t ∈ ( )y t ∈ and ( )u t ∈ are the state, output and control input, respectively; A, b, c are constant real matrices or vectors with appropriate dimensions; τ > 0 and d > 0 stand for the time delay and hysteresis, respectively; The symbol u(t) = −sgnd (y(t)) stands for

C. Lin et al.: Existence Analysis for Limit Cycles of Relay Feedback Systems 429

1 if ( ) or ( ) and ( ) 1

( ) 1 if ( ) or ( ) and ( ) 1 1 1 if ( ) and ( ) 1 or ( ) and ( ) 1.

y t d y t d u tu t y t d y t d u t

y t d u t y t d u t

−

−

− −

− , > , > − = −⎧⎪∈ , < − , < =⎨⎪ − , , = = , = − = −⎩

(2)

Define the switching planes:

,nS c dξ ξ+ =: ∈ : = (3)

,nS c dξ ξ− =: ∈ : = − (4)

and let

nS d c dξ ξ± = ∈ : − < < ,

ndS c dξ ξ− = ∈ : < − ,

ndS c dξ ξ+ = ∈ : > .

For later reference, we introduce some concepts below. An absolutely continuous function x(t) is called a solution to system (1) if it satisfies the equations in (1) almost everywhere. For a solution x(t) to system (1), we say that its trajectory intersects S+ (or S−) with a normal switch, if it traverses the switching plane S+ (or S−) from S± to S+d (or from S± to S−d). The intersecting point is called a traversing point. The instant corresponding to a traversing point is called a traversing instant. We say that a trajectory intersects S+ (or S−) with normal switches, if whenever the trajectory intersects S+ or (S−), it intersects with a normal switch. A set K is said to be convex, if for any s1, s2∈K, we have λ s1 + (1 − λ) s2∈K holds for all λ∈[0,1]. A subset of n

is compact if and only if it is closed and bounded. In the next section, we will study the existence of limit cycles for system (1) using the following fixed point theorem (see, e.g., [13]).

Lemma 2.1. (Brouwer’s fixed point theorem) Let K be a non-empty compact convex subset of

n and let g: K → K be a continuous mapping. Then, g has a fixed point.

III. RESULTS

In this section, we first establish some useful lemmas, and then give the main result for the existence of limit cycles. Let G(s) = c(sI − A)−1b. Then, we have G(0) = −cA−1b if A is invertible.

Lemma 3.1. Suppose that A is Hurwitz and G(0) > d. Then given an initial condition ξ (0) the system trajectory starting from ξ (0) will intersect either S+ or S− in a finite time.

Proof: We prove by contradiction. Suppose that a solution ξ (t) to system (1) starting from the initial condition ξ (0) will hit neither S+ nor S− in a finite time. The trajectory of ξ (t) is governed by

( )00

( ) (0)tAt A t zt e e bu dzξ ξ −= + ∫

10(0) ( )At Ate e I A buξ −= + − ,

where u0 = 1 or −1. Since eAt → 0 as t → + ∞ (due to A Hurwitz), we see that 1

0( ) as .c t cA bu tξ −→ − → +∞ Thus, for some T > τ,

0( ) for 1c t d t T u> , > , =ξ ,

0( ) for 1c t d t T u< − , > , = −ξ ,

which contradicts the control law of system (1). This completes the proof.

Lemma 3.2. Suppose that A is Hurwitz. Let P = PT > 0 be the unique solution to the Lyapunov equation

,TA P PA I+ = − (5)

and define 12

|| || 2 || ||max || ||

PbP

≤=

ξω ξ , (6)

12 || || n PΩ = ∈ : ≤ξ ξ ω . (7)

Then, Ω is invariant for the solutions to system (1), i.e., any solution to system (1) starting from Ω will remain in Ω for all t ≥ 0.

Proof: We prove the invariant property of Ω by contradiction. Suppose there is a solution ξ (t) starting from ξ (t0)∈Ω such that ξ (t2)∉Ω for some t2 > t0, i.e.,

22 2( ) ( ) .T t P tξ ξ ω>

Noting ξ

T(t0) P ξ (t0) ≤ ω2, by continuity of ξ (t) there exists a time t1∈[t0, t2) such that

21 1( ) ( )T t P tξ ξ ω= ,

21 2( ) ( ) ( ]T t P t t t t> , ∀ ∈ ,ξ ξ ω . (8)

This implies that for t∈(t1, t2], we have

430 Asian Journal of Control, Vol. 6, No. 3, September 2004

|| ( ) || 2 || || .t Pbξ > (9)

Now, let V(x) = xTPx. It follows that

( ) ( ) 2T T TdV x x A P PA x x Pbudt

= + +

2 2|| || 2 || || 2 || || || || .Tx x Pbu x Pb x= − + ≤ − +

It is seen that for || x || > 2 || Pb ||, we have ( ) 0V x < . So,

from (9), for t∈(t1, t2], it holds ( ( )) 0dV tdtξ

< , yielding

11( ) ( ) ( ( )) ( ( )) ( ( ))

tTt

t P t V t V t V z dzξ ξ ξ ξ ξ= = + ∫

21( ( ))V tξ ω≤ = ,

which contradicts (8). This completes the proof.

Remark 3.1. Under the conditions of Lemma 3.1, it is sean that the region Ω is also a contraction region in the sense that any trajectory of system (1) will eventually enter Ω. From Lemma 3.2, it is easy to know that there exists a scalar a > 0 such that any solution ξ (t) to system (1) starting from ξ (0)∈ Ω satisfies

|| ( ) || 0.t a tξ ≤ , ∀ ≥ (10)

Besides, under the conditions of Lemma 3.1, the sets SΩ+ and SΩ− are non-empty, where

,S SΩ+ +: = Ω ∩ (11)

.S SΩ− −: = Ω ∩ (12)

This is because any trajectory will traverse S+ and S− in the region Ω.

The next lemma specifies a lower bound of the time between any two traversing instants.

Lemma 3.3. Suppose that A is Hurwitz and G(0) > d. Let h be the time for a trajectory of system (1) to go from SΩ+ (or SΩ−) to SΩ− (or SΩ+). Then, we have h > τ0, where τ0 > 0 is such that

1 0|| ||Ae Iδ τ δ τ− < , ∀ ≤ , (13)

with

1 12 0

|| || ( 3 || ||)d

c a A bτ

−= >

+. (14)

Proof: Without loss of generality, let ξ (0) ∈SΩ+ and the trajectory of ξ (t) evolving from ξ (0) spends time h to reach ξ (h). For some tτ∈[0, h] and uτ = ± 1, we have

( )

0( ) (0)

tAt A t zt e e bu dzττ τ

τ τξ ξ −= + ,∫

( ) ( )

0( ) ( ) ( )

h tA h t A h t zh e t e b u dzττ τ

τ τξ ξ−− − −= + −∫

( )1 1 1( (0) ) 2 .A h tAhe A bu A bu e A buττ τ τξ −− − −= + + −

Since cξ (0) = d and cξ (h) = −d, it follows that

2 ( ) 0d hψ+ = , (15)

where ( )1 1( ) ( )( (0) ) 2 ( ) .A h tAhh c e I A bu c e I A bu−− −= − + − −τ

τ τψ ξ (16)

Now, suppose h ≤ τ0. We show that this will lead to a contradiction. Since h ≤ τ0, we have || eAh − I || < τ1 and || eA(h−tτ) − I || < τ1 in (6). Taking into account || ξ (0) || ≤ a (Remark 3.1), we obtain

1 11 1|| ( ) || || || (|| (0) || || ||) 2 || || || ||

2h c A b c A b

dψ τ ξ τ− −< + +

≤ .

This contradicts (15). Hence, h > τ0. This completes the proof.

Remark 3.2. From the proof of Lemma 3.3, we see that τ0 is independent of the solution ξ (t) and the time delay τ. This means that any trajectory starting from SΩ+ (or SΩ−) will spend a time longer than τ0 to intersect SΩ− (or SΩ+).

Next, we present our main result.

Theorem 3.1. Consider system (1). If

(i) G(0) > d ,

(ii) τ ≤ τ0, and

(iii) A is Hurwitz ,

where τ0 is as in Lemma 3.3, then system (1) has a limit cycle.

Proof: From Lemma 3.1, any trajectory x(t) of system (1) starting from x(0)∈Ω will intersect, without loss of generality, S+. Let the traversing point be x(t1)∈S+ (at time t1). Since τ ≤ τ0, by virtue of Lemma 3.3, after switching, the trajectory of x(t) will intersect and traverse S− at some point x(t2) (at time t2 > t1 + τ0), and then intersect and traverse S+ again at some point x(t3) (at time t3 > t2 + τ0). Define the mapping Φ: SΩ+ → SΩ+ as

1 3( ( )) ( ).x t x tΦ = (17)

Then Φ maps SΩ+ into itself. The continuity of Φ can be shown from the expressions of solutions to system (1). The fact that SΩ+ is non-empty is obvious and the compactness of SΩ+ can be easily verified from the

C. Lin et al.: Existence Analysis for Limit Cycles of Relay Feedback Systems 431

definition for compact set. For the convexity, it is seen from the following

1 2 Sξ ξ Ω+∀ , ∈

12

1 2 1 2

1 2 1 2

( (1 ) ) (1 )

|| ( (1 ) ) || (1 ) ,

c d S

P

λξ λ ξ λξ λ ξ

λξ λ ξ ω λξ λ ξ++ − = ⇒ + − ∈⎧⎪⇒ ⎨

+ − ≤ ⇒ + − ∈ Ω⎪⎩

where λ∈[0,1]. Hence, by Lemma 2.1, there exists x*∈ SΩ+ such that Φ (x*) = x*. This shows that x*(t) starting from x* is a limit cycle. This completes the proof.

Remark 3.3. The result of Theorem 3.1 is for systems of the form (1) with d > 0 and τ > 0. For systems with no delay and hysteresis (i.e., d = 0 and τ = 0), fast switches may occur, and there are possibly infinite (but countable) number of switching times during a finite time interval. (See Theorem 1 in [5].) In case of this phenomenon, the analysis is very hard. As for the case of d > 0 and τ = 0, under the conditions of Theorem 3.1 and the well-posedness assumption, the result in Theorem 3.1 is still valid (by removing item (ii)).

Remark 3.4. It is easy to show that under some constraint, Theorem 3.1 can be made valid for systems with any τ > 0. For instance, if in the region Ω the time duration between any two successive traversing instants is greater than τ, then Theorem 3.1 still holds by removing item (ii).

IV. CONCLUSION AND DISCUSSION

This paper gives a sufficient condition for the existence of limit cycles of linear systems under relay feedback. However, there are still lots of work need to be done. To establish a more easy-checking condition is a future work.

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